TEICHMULLER SPACES AND GEOMETRY OF MOBIUS TRANSFORMATIONS

TEICHMULLER 空间和莫比乌斯变换的几何

• 批准号: 11640162
• 负责人: OKUMURA Yoshihide
• 金额: $ 2.24万
• 依托单位: SHIZUOKA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640162/
• 关键词: Teichmuller space; discrete group; Riemann surface; simple dividing loop; Teichmuller modular group; Mobius transformation; byperbolic manifold; real analytic manifold; 不連続群; 大域座標

My research during this term consists mainly of the following three branches :1. Consideration of the relation between angle parameters and the geometry of Mobius transformations.2. Representation of the Teichmuller modular groups (the mapping class groups) by angle parameters.3. Characterization of simple dividing loops on Riemann surfaces analytically.In order to obtain global real analytic and simple representations of the Teichmuller spaces, I introduced new angle parameters. I showed that the Theichmuller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmuller spaces.Considering the axes of the generators and these products of Fuchsian groups, for example the once-holed torus Fuchsian groups, I found out the high symmetry of the arrangement of these axes. I investigated the relation among such geometry of Mobius transformations, traces and angle parameters, using the one-half powers of Mobius transformations an … More d the hyperbolic geometry. From these observations, the much relation and information of angle parameters were obtained. From such information of angle parameters, I tried to represent the Teichmuller modular groups by only angle parameters. I considered the following :(1) Relation between angle parameters and length parameters representing these groups.(2) Concrete description of such groups by only angle parameters in the cases that it is easy to calculate.(3) Choice of angle parameters (inductively) that simply represent the Theichmuller modular groups of the general cases.I especially studied the representation of the Theichmuller modular groups of a once-holed torus and a compact Riemann surface of genus 2.Furthermore, I gave the necessary and sufficient condition of a simple loop L on a Riemann surface S to be dividing, using the lifts of a Fuchsian group G representing S to the special linear group SL (2, C). For example, if S is a compact Riemann surface of genus p (>1), then the following is obtained :The number of the lifts of G is 2 to the 2p-th power. Let g be an element of G corresponding to L.Then L is to be dividing if and only if for any lift of G, the matrix corresponding to g always has the negative trace. Less

我这学期的研究主要包括以下三个方面：1.考虑角度参数与Mobius变换几何关系. Teichmuller模群（映射类群）的角参数表示. Riemann曲面上简单分割环的解析表征为了获得Teichmuller空间的全局真实的解析和简单表示，我引入了新的角度参数。证明了Teichmuller空间只用角参数来描述，这使得分析典型的Teichmuller空间的角参数空间变得容易，考虑到Fuchsian群的生成元和这些乘积的轴，例如一次有孔环面Fuchsian群，我发现这些轴的排列具有高度对称性。本文利用Mobius变换的二分之一幂， ...更多信息 双曲几何学。从这些观测中，获得了大量的角度参数之间的关系和信息。从这样的角度参数的信息，我试图表示Teichmuller模群的角度参数。考虑了以下问题：（1）表示这些群的角度参数和长度参数之间的关系。(2)在易于计算的情况下，仅用角度参数对这类群进行具体描述。(3)角度参数的选择（归纳地）简单表示一般情况下的Theichmuller模群，特别研究了亏格为2的单孔环面和紧致Riemann曲面的Theichmuller模群的表示，并给出了Riemann曲面S上的单圈L可除的充要条件，利用表示S的Fuchsian群G到特殊线性群SL（2，C）的提升。例如，如果S是亏格为p（> 1）的紧致黎曼曲面，则得到：G的提升数为2的2p次方。设g是G中对应于L的元素，则L是可除的当且仅当对G的任何提升，g对应的矩阵总是有负迹。少

项目成果

Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type, II"J.Math.Soc.Japan. 53(to appear). (2001)

Hiroki Sato：“实型经典肖特基群的约根森不等式，II”J.Math.Soc.Japan。

Toshihiro Nakanishi: "Areas of two-dimensional moduli spaces"Proc.Amer.Math.Soc.. (印刷中).

Toshihiro Nakanishi：“二维模空间的面积”Proc.Amer.Math.Soc..（正在出版）。

Yoshihide Okumura: "Angle parameters for Teichmuller spaces and its application"RIMS Kokyuroku, Research Institute for Mathematical Sciences Kyoto Univ. (to appear). (2001)

Yoshihide Okumura：“Teichmuller 空间的角度参数及其应用”RIMS Kokyuroku，京都大学数学科学研究所。

Yoshihide Okumura: "Lifting problem and its application to Riemann surfaces"Eighth International Conference on Complex Analysis. (印刷中). (2001)

Yoshihide Okumura：“提升问题及其在黎曼曲面中的应用”第八届国际复分析会议（2001 年）。

Hiroki Sato: "One-parameter families of extreme discrete groups for Jorgensen's inequality"Contemporary Math. (The First Ahlfors-Bers Colloquium). Vol.256. 271-287 (2000)

Hiroki Sato：“乔根森不等式的极端离散群的单参数族”当代数学。